Residual properties of the modular group and other free products

Using a probabilistic approach we establish new residual properties of the modular group PSL2(Z), and of more general free products. We prove that the modular group is residually in any infinite collection of finite simple groups not containing a Suzuki group Sz(q) or a 4dimensional symplectic group PSp4(q) with q a power of 2 or 3. This result is best possible, since the groups excluded are not quotients of the modular group. We also show that if S is a collection of classical groups of unbounded rank, then an arbitrary free product A ∗ B of nontrivial finite groups, not both 2-groups, is residually S, and prove results about free products A ∗ Z.